Surface Area of Cylinders

(1) Right Circular Cylinder:

  1. Curved surface area = perimeter x height of cylinder i.e. S = 2\pi rh
  2. Area of each of the flat surface, i.e., of ends i.e.  = \pi {r^2}
  3. Total surface area  = 2\pi rh + 2\pi {r^2} =       2\pi r\left( {r + h} \right)

Example:

Find the height of the solid circular cylinder if total surface area is 600 sq.cm and radius is 5cm.

Solution:
            Here    r = 5cm
            Total surface area  = 2\pi {r^2} + 2\pi rh = 660
            Or        2\pi  r\left( {r + h} \right) = 660
            Or        2  \times \frac{{22}}{7} \times 5\left( {5 + h} \right) = 660
            Or        5 + h =  \frac{{660 \times 7}}{{5 \times 44}} = 21
            \therefore           h = 16cm

Example:

A cylinder vessel, without lid, has to be in coated on both its sides. If the radius of its base is \frac{1}{2}m and its height is 1.4m, calculate the cost of tin coating at the rate of Dollar 2.25per 1000 sq.cm.

Solution:
            Given that
            Radius of the base of cylindrical vessel, r =  \frac{1}{2}m = 50cm
            Height,              h = 1.4m = 140m
            \therefore Area to be tin coated  = 2\left( {{\text{curved surface + area  of base}}} \right)
                                                    = 2\left(  {2\pi rh + \pi {r^2}} \right)
                                                    = 2\pi r\left(  {2h + r} \right)
                                                    = 2  \times 3.14 \times 50\left( {2 \times 140 \times 50} \right) = 314 \times 330
                                                    = 103620sq.cm
            Now, cost of tin coating per 1000sq.cm = Dollars 2.25
            \therefore  Total cost of tin coating  =  \frac{{2.25}}{{1000}} = 103620 = 233.15 Dollars

(2) Hollow Circular Cylinder:

  1. Curved surface area  = 2\pi Rh + 2\pi rh = 2\pi       \left( {R + r} \right)h
  2. Total surface area  = 2\pi \left( {R + r}       \right)h + 2\pi \left( {{R^2} - {r^2}} \right)

(3) Elliptic Cylinder:

The cylinder with a base which is an ellipse is called an elliptic cylinder. If a and b are the semi-major axis and semi-minor axis and his the height, then

  1. Volume  = \pi abh
  2. Curved surface area  = \pi \left( {a + b}       \right)h
  3. Total surface area = \pi \left( {a + b}       \right)h + 2\pi ab

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